A cooling channel cover of the generic type is known from EP 1 238 191 B1. This known cooling channel cover has an intrinsically elastic supply element which is received in an opening provided in the cooling channel cover and fastened thereto by means of a latching connection or by clipping in. For mounting purposes, the known supply element is deformed elastically inward in order to be able to guide it through the opening in the cooling channel cover. This requires that the known supply element is provided only with very small solid latching noses and bearing surfaces which have only very little contact with the cooling channel cover. Reliable operation of a piston provided with such a cooling channel cover is not reliably ensured on account of the forces occurring on the latching noses and bearing surfaces during engine operation and on account of the thus occurring wear in the region of the latching noses and bearing surfaces.